After renaming my PC I activated my bundle with a new serial number. SQL Compare accepted the new number and works.

SQL Prompt, however, does not work.

It allows me to enter the new serial number, claims that the whole bundle has been activated, but it remains disabled. WHen I reopen Enter Serial Number dialog box, I still see the old serial number (with a green check mark). I have repeated this process twice.

The computer rename probably changed the information that the licensing uses to identify a unique computer. Deleting the licence file must have reset the machine hash. I'm glad it seems to be working now.